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d5b562f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingFreeStep
namespace OAI
/-! # The two exact energy components on an expanding torus
The normalized Fourier space has a fixed Hilbert norm. These two bounded
observations recover the low and high terms of the radius-dependent norm.
-/
open scoped ENNReal
namespace DefocusingNLS
noncomputable def expandingLowFraction (a k L : ℝ) (n : frequencyLattice) : ℝ :=
L ^ (2 * a) * (1 + ‖n‖ ^ 2) ^ (6 - a) / expandingSobolevWeightSq a k L n
theorem expandingLowFraction_bounds (a k L : ℝ) (hL : 1 ≤ L)
(n : frequencyLattice) : 0 ≤ expandingLowFraction a k L n ∧
expandingLowFraction a k L n ≤ 1 := by
constructor
· exact div_nonneg (by positivity) (expandingSobolevWeightSq_pos a k L hL n).le
· apply (div_le_one (expandingSobolevWeightSq_pos a k L hL n)).mpr
unfold expandingSobolevWeightSq
exact le_add_of_nonneg_right (by positivity)
theorem expandingHighFraction_eq (a k L : ℝ) (hL : 1 ≤ L)
(n : frequencyLattice) :
1 - expandingLowFraction a k L n =
L ^ (12 - 2 * k) * ‖n‖ ^ (2 * k) / expandingSobolevWeightSq a k L n := by
have hp := (expandingSobolevWeightSq_pos a k L hL n).ne'
unfold expandingLowFraction
field_simp
unfold expandingSobolevWeightSq
ring
noncomputable def fourierEnergyComponent (w : frequencyLattice → ℝ)
(hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) : FourierL2 →L[ℂ] FourierL2 :=
let F : FourierL2 → FourierL2 := fun f =>
⟨fun n => (Real.sqrt (w n) : ℂ) * f n, by
apply (lp.memℓp f).mono'
intro n
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_nonneg (Real.sqrt_nonneg _)]
exact mul_le_of_le_one_left (norm_nonneg _) (Real.sqrt_le_one.mpr (hw n).2)⟩
LinearMap.mkContinuous
{ toFun := F
map_add' := by intro f g; ext n; exact mul_add _ _ _
map_smul' := by intro c f; ext n; change _ * (c * _) = c * (_ * _); ring }
1 (by
intro f
rw [one_mul]
apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
intro n
change ‖(Real.sqrt (w n) : ℂ) * f n‖ ≤ ‖f n‖
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_nonneg (Real.sqrt_nonneg _)]
exact mul_le_of_le_one_left (norm_nonneg _) (Real.sqrt_le_one.mpr (hw n).2))
@[simp] theorem fourierEnergyComponent_apply (w : frequencyLattice → ℝ)
(hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) (f : FourierL2) (n : frequencyLattice) :
fourierEnergyComponent w hw f n = (Real.sqrt (w n) : ℂ) * f n := rfl
theorem fourierEnergyComponent_norm_sq (w : frequencyLattice → ℝ)
(hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) (f : FourierL2) :
‖fourierEnergyComponent w hw f‖ ^ 2 = ∑' n, w n * ‖f n‖ ^ 2 := by
have he := lp.norm_rpow_eq_tsum (p := 2) (by norm_num)
(fourierEnergyComponent w hw f)
simp only [ENNReal.toReal_ofNat, Real.rpow_two] at he
rw [he]
apply tsum_congr
intro n
rw [fourierEnergyComponent_apply, norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_nonneg (Real.sqrt_nonneg _), mul_pow, Real.sq_sqrt (hw n).1]
theorem summable_fourierEnergyComponent (w : frequencyLattice → ℝ)
(hw : ∀ n, 0 ≤ w n ∧ w n ≤ 1) (f : FourierL2) :
Summable (fun n => w n * ‖f n‖ ^ 2) := by
have h := lp.hasSum_norm (p := 2) (by norm_num) (fourierEnergyComponent w hw f)
simp only [ENNReal.toReal_ofNat, Real.rpow_two] at h
convert h.summable using 1
funext n
rw [fourierEnergyComponent_apply, norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_nonneg (Real.sqrt_nonneg _), mul_pow, Real.sq_sqrt (hw n).1]
noncomputable def expandingLowEnergy (a k L : ℝ) (hL : 1 ≤ L) :
FourierL2 →L[ℂ] FourierL2 :=
fourierEnergyComponent (expandingLowFraction a k L) (expandingLowFraction_bounds a k L hL)
noncomputable def expandingHighEnergy (a k L : ℝ) (hL : 1 ≤ L) :
FourierL2 →L[ℂ] FourierL2 :=
fourierEnergyComponent (fun n => 1 - expandingLowFraction a k L n)
(fun n => ⟨sub_nonneg.mpr (expandingLowFraction_bounds a k L hL n).2,
sub_le_self _ (expandingLowFraction_bounds a k L hL n).1⟩)
theorem expandingEnergy_norm_sq (a k L : ℝ) (hL : 1 ≤ L) (f : FourierL2) :
‖f‖ ^ 2 = ‖expandingLowEnergy a k L hL f‖ ^ 2 +
‖expandingHighEnergy a k L hL f‖ ^ 2 := by
have hl := summable_fourierEnergyComponent (expandingLowFraction a k L)
(expandingLowFraction_bounds a k L hL) f
have hh := summable_fourierEnergyComponent (fun n => 1 - expandingLowFraction a k L n)
(fun n => ⟨sub_nonneg.mpr (expandingLowFraction_bounds a k L hL n).2,
sub_le_self _ (expandingLowFraction_bounds a k L hL n).1⟩) f
rw [expandingLowEnergy, expandingHighEnergy, fourierEnergyComponent_norm_sq,
fourierEnergyComponent_norm_sq, ← hl.tsum_add hh]
have he := lp.norm_rpow_eq_tsum (p := 2) (by norm_num) f
simp only [ENNReal.toReal_ofNat, Real.rpow_two] at he
rw [he]
apply tsum_congr
intro n
ring
end DefocusingNLS
end OAI
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